# What is the statement of the Stokes theorem

## Stokes' theorem

In this article, the **Stokes' theorem** treated. First the **general Stokes' theorem** formulated before briefly on its special cases **Main theorem of differential and integral calculus (HDI)** as well as the **Gaussian integral theorem** is received. In addition, the **classic integral theorem from Stokes** as a further special case of the general will be examined in more detail. Finally, the calculation takes place **two examples**.

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- Stokes general integral theoremin the text
- Underlying topological principlein the text
- Stokes' theorem as a classical Stokes integral theoremin the text
- in the text

### Stokes general integral theorem

If from **Stokes' theorem** is in question, in most cases it is the **classic Stokes integral theorem** meant. It represents a special case of the **general integral theorem of Stokes** which reads as follows:

Be open and a **oriented -dimensional submanifold** With as a **continuously differentiable -Shape** in . Then applies to everyone **compact amount with a smooth edge**

,

in which the induced orientation carries and the **outer derivative** of designated.

### Underlying topological principle

The **Stokes' theorem** lies that **topological principle** is based on the fact that when paving a piece of land by identically oriented "paving stones" the **inner ways** in **opposite direction** will go through, which leads to their **Mutually cancel contributions to the line integral** and only that **Contribution of the boundary curve** remains.

### Main theorem of differential and integral calculus as a special case

For **degenerate** the **general integral theorem of Stokes** to the **Law of differential and integral calculus**: Be an open interval and a continuously differentiable function. Then the following applies:

### Gauss integral theorem as a special case

Another special case follows from the **general integral theorem of Stokes** the **Gaussian integral theorem**. To show that will be chosen and be it

,

i.e. with the continuously differentiable vector field . The roof shows over indicates that this factor must be omitted. Be also the **outer unit normal field**, then applies

With also results

Ultimately, this gives that **Gaussian integral theorem**

### Stokes' theorem as a classical Stokes integral theorem

Often, and especially in technical courses and physics, there is talk of **Stokes' theorem**. This is usually the **classic integral theorem from Stokes** meant which one too **Kelvin-Stokes theorem** or **Rotation set** is called. Together with the **Gaussian integral theorem** he plays an essential role in the formulation of the **Maxwell's equations in the integral form**.

### Special case of Stokes' general integral theorem

The **classic theorem of Stokes** results like that **HDI** and the **Gaussian integral theorem** as **Special case of Stokes' general integral theorem**. In this case the **open amount** as well as that **continuously differentiable vector field** considered. put one **two-dimensional submanifold** whose orientation through the **Unit normal field** is given. On the submanifold be on **Compact form** given which one **smooth edge** own. This in turn is through the **Unit tangent field** oriented. With the in **continuously differentiable Pfaffian form**

and

thus results in the **Stokes' theorem**:

In another notation it reads:

### Stokes' theorem

The following can be seen:

The **Stokes' theorem** states that a **Area integral** about the **rotation** one **Vector field** under certain conditions in a **closed curve integral** over to the curve **tangential component** of **Vector field** can be converted. The curve traversed must be the **Edge of the area under consideration** correspond.

### Stokes' theorem

In the following, the **Stokes' theorem** be proven. For this proof, however, there is a small one **condition** to the surface posed. This should be the **graph** a function be which over one **area** in the Level is defined. With and be the **Projections** of and the **counterclockwise edge** on the -Level. be through

parameterized, from which with the help of the **Chain rule** follows:

For the im **Stokes' theorem** looked at **Curve integral**:

By summarizing it results:

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